Hi, what is A in 1). Is it a matrix, linear map, ..? If yes, the answer is positive and you need only to show that for any
you can use the same argument for 2)
Two questions am not understanding?
1) It says;
Are the foloowing subspaces?Support your answer with detials.
U1={X belong to R^n:AX=3X}
and
2)
We fix a non-zero vector d belonging to R^3, and define U3={X beloninig to R^3: proj(d)(X)=0}, where projd(x) is the projection of X onto d
To show that S is a subspace of , it suffice to show it's stable by addition and multiplication by a scalar (real number). So you need to prove that has these properties using the fact that and i.e. A is linear. Also multiplication by 3 is linear.
for 2), proj(d) is also linear.
Projecting of X on a vector d is equivalent to taking the scalar product X.d and form the vector (X.d)d/(d^2) i.e. proj(d)(X) = (X.d)d/(d^2) (d^2 is the square of the norm of d). You should know that scalar product is linear i.e. (X+Y).d = X.d+Y.d and (cX).d = c(X.d)